5 research outputs found
Signature Gr\"obner bases in free algebras over rings
We generalize signature Gr\"obner bases, previously studied in the free
algebra over a field or polynomial rings over a ring, to ideals in the mixed
algebra where is a principal
ideal domain. We give an algorithm for computing them, combining elements from
the theory of commutative and noncommutative (signature) Gr\"obner bases, and
prove its correctness.
Applications include extensions of the free algebra with commutative
variables, e.g., for homogenization purposes or for performing ideal theoretic
operations such as intersections, and computations over as
universal proofs over fields of arbitrary characteristic.
By extending the signature cover criterion to our setting, our algorithm also
lifts some technical restrictions from previous noncommutative signature-based
algorithms, now allowing, e.g., elimination orderings. We provide a prototype
implementation for the case when is a field, and show that our algorithm
for the mixed algebra is more efficient than classical approaches using
existing algorithms.Comment: 10 page
Monomial orderings, rewriting systems, and Gröbner bases for the commutator ideal of a free algebra
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum